An efficient algorithm for finding a two-pair, and its applications

Let G = (V, E) be an undirected, finite, and simple graph. All the definitions not given here may be found in [l]. A pair (u, o) of nonadjacent vertices in G form a two-pair if every chordless path between u and v is of length two. We shall digress briefly to remark on the applications of our problem. One of the fundamental notions in computational geometry is the concept of visibility. A visibility graph has vertices which correspond to geometric components such as points or lines, and edges which correspond to visibility of these components from each other. O’Rourke [5] declares that the fundamental problems involving visibility in computational geometry will not be solved until the combinatorial structure of the visibility graphs is more fully understood. There is an intimate connection between the visibility problem and the polygon covering problem. Recently, several interesting structural characterizations of visibility graphs for regions inside a simple orthogonal polygon were reported in [3, 41. In particular, the visibility graph of a special type of orthogonal polygon called Class 3 polygon [3] is a weakly triangulated graph. It is also shown in [3, 41 that the polygon covering problem for the Class 3 polygon reduces to the clique covering problem on weakly triangulated graphs. The fact that weakly triangulated graphs are perfect is used to derive several insightful duality relationships for Class 3 polygon.